A mathematical puzzle that stumped Paul Erdős for decades has finally been solved by a computer.
April 29, 2026
Original Paper
Size-4 Counterexamples to the Sidon-Extension Conjecture
arXiv · 2604.25214
The Takeaway
The Sidon-extension conjecture was a long-standing question about how specific sets of numbers can be expanded. This paper provides the first definitive counterexamples to the theory, using size-4 sets to disprove the claim. The researchers used a combination of empirical search and formal logic to settle a problem posed by one of history's most prolific mathematicians. Disproving this conjecture changes the understanding of how perfect difference sets are structured. This result demonstrates that even the most famous mathematical intuitions can be overturned by precise computational analysis.
From the abstract
A finite set $S \subset \mathbb{Z}$ is a \emph{Sidon set} if its pairwise differences are distinct. A \emph{perfect difference set} (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n + 1$) of size $n$ such that every nonzero residue arises exactly once as a difference of two elements of $B$. Erdős's \$1000 conjecture -- that every finite Sidon set extends to a finite PDS -- was disproved by Alexeev and Mixon (arXiv:2510.19804, October 2025) via the size-5 counterexamples $\{1,2,4